Is there a way to legally get siblings to help pay for caregiving?

It takes two things to receive Medicaid payment for a NH bed. Medical need and financial need.
Medical need is a sign off by his doctor. Financial need is confirmed by completing an application that includes the last five years financial records and a review of all assets.
You could probably visit the NH of your choice and start there. Each state is a little different on requirements.
Your local area agency on aging could be contacted for a needs assessment and tell you what services are available for your dad.
If your siblings contributing funds would enable you to keep him home and that is your and his desire, why not meet with the siblings and let them know what it would cost each of them to continue that care?
Is it possible that your father is ready for hospice? Those services could be looked into as well. With a prognosis your siblings might be better able to determine their ability to contribute.
Also, should your father be a veteran he might be eligible for veterans aid and attendance assistance.
I’m sorry he is so ill. I know you are tired.
Make a few phone calls tomorrow and see if you can get things moving to get yourself some help to care for your dad.

It might be an easier route directly from a hospital, bc they have social workers to help you navigate the system, but if you can handle all else, I think it's just a requirement to have the dr sign off. Medicaid, however, I know nothing about.

No , there is no legal way to make your siblings pay. Your best move would be to get him in to the hospital. I don't think there are very many admissions into a nursing home without being in hospital first. Paying patient or medicaid patient, if you refuse to take him back home, the hospital will get him into a nursing home.